Solution

Our solution concept (formally defined below) requires that the level of violence chosen in each period maximizes the opposition expected payoff for the rest of the game given the future equilibrium violence choices.Footnote ¹¹ To compute the optimal violence choice in period t, the party in opposition has to compare the expected payoff for the remainder of the game entering the next period as incumbent or opposition. In the last period, this is relatively straightforward, as the values of ending the game in either role are the exogenously given π* _I (T + 1) and π* _O (T + 1). So, the payoff for the party that is the opposition in the final period as a function of their violence choice is

\fontsize{9.3}{11.3} \selectfont{$$\begin{eqnarray*} &&\underbrace{\sum _{t=1}^{T-1} \delta ^{t-1} u_t}_{\text{past payoffs}} - \underbrace{\delta ^{T-1} c(v_T)}_{\text{period $T$ payoff}} \\ &&\quad+\, \underbrace{\delta ^T \lbrace p(v_T; k_T) \pi _I^{*}(T+1) + [1 - p(v_T; k_T)] \pi _O^{*}(T+1)\rbrace }_{\text{continuation value}} \end{eqnarray*}$$}

Since the past payoffs are fixed, when characterizing the optimal level of violence it is simpler to work with the net present value of the game starting in period t, which only considers the payoffs beginning in period t and normalizes by the discount rate. So, for the last period this is given byFootnote ¹²

\begin{eqnarray*} \pi _O(T, v_T) &=& -c(v_T) + \delta \lbrace p(v_T; k_T) \pi _I^{*}(T+1)\\ &&+\, [1 - p(v_T; k_T)] \pi _O^{*}(T+1)\rbrace \end{eqnarray*}

To be sequentially rational, the violence level in the last period must maximize π _O (T, v_T ). Taking the first order condition, an interior optimal violence choice in the v_T solves

(1) $$\begin{eqnarray} \underbrace{\frac{\partial\, p(v_T; k_T)}{\partial v_T}}_{\Delta\text{Pr(take over)}} \underbrace{\delta [\pi _I^{*}(T+1) - \pi _O^{*}(T+1)]}_{\text{discounted value of taking over}} &= c^\prime (v_T). \end{eqnarray}$$

The left-hand side of equation (1) is the marginal benefit to committing more violence in period T, which is equal to the change in the probability of ending the game as the incumbent times the discounted difference between ending the game as the incumbent versus as opposition. The right-hand side of this equation is the marginal cost to committing more violence. Since there are diminishing returns to committing more violence and an increasing marginal cost, this equation has a unique solution v* _T where the marginal benefit equals the marginal cost.Footnote ¹³

The optimal violence choice in every period has this general form: the opposition commits the level of violence where the marginal cost of committing more violence equals the marginal benefit via increasing the chance of taking over office in the next period. The only complication—which will drive the intertemporal substitution effects we aim to model—is that the benefit of taking over office in the next period depends on the future effectiveness of violence. However, once we know the violence level in period T (derived above), we can compute the net present value of entering period T as incumbent versus opposition, which gives the optimal level of violence in period T − 1. This gives the net present value of entering the next to last period in each role, and hence the optimal violence level in period T − 2, etc.

Formally, the opposition objective function (π _O (t, v_t )), optimal violence choice (v* _t ), and the net present values for entering period t in each role (π* _O (t) and π* _I (t)) can be defined recursively as

(2) $$\begin{eqnarray} \pi _O(t, v_t) &=& c(v_t) + \delta \lbrace p(v_t; k_t) \pi _I^{*}(t+1) \nonumber\\ &&+\, [1 - p(v_t; k_t)] \pi _O^{*}(t+1)\rbrace , \end{eqnarray}$$

(3) $$\begin{eqnarray} v_t^{*} \in \arg \max_{\hspace*{-20pt}v_{t} \ge 0} \pi _O(t, v_t), \end{eqnarray}$$

(4) $$\begin{eqnarray} \pi _O^{*}(t) = \pi _O(t, v_t^{*}), \end{eqnarray}$$

(5) $$\begin{eqnarray} \pi _I^{*}(t) &=& \psi\, + \delta \{ p(v_t^{*}; k_t) \pi _O^{*}(t+1) \nonumber\\ &&+\, [1 - p(v_t^{*}; k_t)] \pi _I^{*}(t+1)\}. \end{eqnarray}$$

An equilibrium to the model is characterized by sequences v* _t , π* _O (t), and π* _I (t) that solve equations (2)–(5) for t = 1, . . ., T. It is also convenient to write the difference between the net present value of entering period t as the incumbent versus the as the opposition as Δ* _t ≡ π* _I (t) − π* _O (t). So, the first order condition for an interior level of violence v_t is

(6) $$\begin{eqnarray} \delta \frac{\partial\, p(v_{t}; k_{t})}{\partial v_{t}} \Delta ^{*}_{t+1} = c^\prime (v_t). \end{eqnarray}$$

As long as Δ* _{t + 1} > 0—which means it is better to enter the next period as the incumbent rather than as the opposition—this optimization problem has a unique interior solution v* _t . If it is better to enter period t + 1 as the opposition rather than as the incumbent, there is no benefit to violence and hence it is optimal to pick v* _t = 0.Footnote ¹⁴ So:

Proof. Follows from the above analysis; see the Appendix for details. $\Box$

Next we analyze how changing the effectiveness of violence in a given period affects the pattern of violence throughout the game. For simplicity we only consider the plausibly realistic case where it is better to enter as the incumbent in every period (i.e., when Δ* _t > 0 for all t); see the Online Appendix for discussion of how loosening this restriction affects the results.

The direct effect of increasing the effectiveness of violence in period t is that by increasing the marginal effect of v_t on taking over office, the marginal benefit to violence increases, leading to a higher choice in period t. Formally, implicitly differentiating equation (6) gives that $\frac{\partial v_t^{*}}{\partial k_t} > 0$ .

To formalize the indirect effect, consider how making violence more effective in future periods (i.e., t′ > t) affects behavior in period t. (By sequential rationality, violence levels in previous periods t′ < t do not affect the choice in period t.) Referring back to equation (6), this will depend on how the future effectiveness of violence changes the relative value of entering the next period as the incumbent (Δ* _{t + 1}). When t′ = t + 1, the result is straightforward: making violence more effective in the next period unambiguously makes the opposition better off and the incumbent worse off entering this period. This reduces the relative value of entering the next period as the incumbent, so there is less violence in period t. For periods t′ > t + 1 the technical analysis is more subtle, but for all of the examples we present in our illustrations, reducing the effectiveness of violence at any future period decreases the level of violence today (see the Online Appendix for examples where this does not hold). Intuitively, as long as it is better to enter each period as the incumbent, and taking control of the government by force is rare, anything that makes doing so easier in the future reduces incentives to commit violence today. Formally:

Proof. See the Appendix. $\Box$

While the model has not yet explicitly addressed elections, this result formalizes the central intuition about the direct and indirect effects of making violence more effective in a given period. The obvious direct effect is that there is more violence in this period. However, making violence more effective in period t has an indirect effect of decreasing violence in periods t − 1, and potentially in all periods prior to period t.

The idea that political actors time their actions is fairly well documented, especially in the terrorism literature. For instance, the Boston Marathon bombers had initially planned their attack for the July 4 festivities (Boston Reference Boston2013), and the so-called “underwear bomber” had timed his attack for Christmas Day (Hudson Reference Hudson2013). Such violent actors are also known to specifically wait for elections so that their attacks would have a greater impact. A recent prominent example of such an approach is the the 2004 train bombings in Madrid that took place three days before the Spanish voters went to the ballot, and is now known to have been timed specifically to influence the election outcome (Richburg Reference Richburg2004). Other similar instances include the Kurdistan Workers’ Party (PKK) attacking Turkish security forces to sway the electorate in April 2015 (Aksoy Reference Aksoy2015), and intelligence reports indicating that Pakistani terrorists planning suicide attacks timed for the Indian elections (Shukla Reference Shukla2014). Violence (including but not limited to terrorism) may also be more prevalent during seasons when it is easier to recruit fighters, for example because wages are lower outside of harvest seasons (Guardado and Pennings Reference Guardado and Pennings2015).Footnote ¹⁵

Proposition 2 indicates that this intuition can apply to the incentives to commit violence to gain political power. While some of these examples are from groups with no stated aspiration to actually take control of the government, in Section 4 we further argue that the logic applies to a wide class of political violence committed by various actors and various aims.

A Simple Example

Suppose in a society without elections, the effectiveness of violence is k_t = k_n for all periods. In a society with elections, the effectiveness of violence is k_e > k_n in electoral periods and k_n for other periods.

A simple microfoundation for this assumption is that the opposition can commit “nonelectoral” violence in any period, but electoral violence is only possible (or only useful) in electoral periods. So, in electoral periods, the opposition—and, as formalized in Section 4, other actors—have multiple ways to influence political outcomes with violence. While electoral violence can look much different than other forms of violence, for our purposes adding a new violent technology during elections is equivalent to assuming that violence is more effective in electoral periods (see the Online Appendix for a formal statement of this point). More broadly, we make this seemingly unusual assumption because our primary goal is to show how elections can reduce violence even if fighting peaks around elections. So this assumption constitutes a “hard case” for our aims.Footnote ¹⁶

As demonstrated by Proposition 2, introducing elections in this manner has a direct effect of increasing the level of violence in electoral periods but also an indirect effect of decreasing violence in the periods before elections, and potentially all nonelectoral periods. To show that either effect can be larger, we present several illustrative cases. For all of these examples, c(v_t ) = v ² _t and p(v_t ; k_t ) = k_t [1 − (1 + v_t )⁻¹].

Figure 3 presents a comparison of violence levels in two societies that only vary in the presence of elections. In both societies, there are 20 periods, and the effectiveness of violence in nonelectoral periods is k_n = 0.3. Without elections (the grey line), all periods are nonelectoral, and the equilibrium violence choice is constant.Footnote ¹⁷ In the second society (black curve), there are elections in periods 6, 12, and 18 (the vertical lines), where the effectiveness of violence increases to k_e = 1. The horizontal black line represents the average violence level with elections.

FIGURE 3.

Comparison of Levels of Violence with No Elections (grey line) and Elections (black line)

The black curve presents a similar (if tidier) pattern as shown in the cases of Kenya and Ghana in Figure 1 with spikes in levels of violence in electoral periods (or years). While we cannot directly know what the counterfactual level of violence would be without elections using observational data, with the theoretical model we can make this comparison by looking at violence levels in an otherwise identical society with no elections. In this case, the black line is below the grey line, indicating that the spikes in electoral periods are more than offset by the decrease in violence in nonelectoral periods, leading to less fighting on average.Footnote ¹⁸ This example illustrates the first main result of the article in a simple fashion: the existence of a political violence cycle does not imply societies would become more peaceful without elections.Footnote ¹⁹

Next, we consider how general this example is by examining how changing various parameters of the model affects patterns and aggregate levels of violence. Unfortunately, analytic results on this relationship in the baseline model prove very difficult (stronger results along these lines emerge in the model with nonviolent actions in the next section). However, it is straightforward to simulate the model for a wide range of parameters and functional forms for the c and p functions to explore how this affects average levels of violence. The Online Appendix contains an extensive analysis of how changing individual parameters affects violence levels for randomly drawn values of the other parameters. In the main text, we focus on several results from this exercise that are of substantive interest and are generally robust to varying other parameters.

Consequential/Competitive Elections

First, Figure 4 shows a counterintuitive and empirically relevant example of how the degree to which violence is more effective during elections affects the average level of violence. The left panel illustrates the per period violence levels with increasingly dark lines as k_e increases. Increasing the effectiveness of violence in the electoral periods leads to bigger spikes of violence during the election, but also larger decreases in violence in nonelectoral periods.

FIGURE 4.

The Effect of Making Elections more Consequential on Average Levels of Violence

The right panel shows how this affects the average levels of violence (solid line) and the difference between the average level of violence in electoral periods versus nonelectoral periods (dotted line). While making violence slightly more effective in electoral periods increases the average amount of violence, making violence much more effective leads to less violence overall. However, increasing the effectiveness of violence unambiguously increases the difference between violence levels in electoral versus nonelectoral periods. So, for part of the parameter space, making electoral period violence more effective leads to bigger spikes of violence in electoral periods but less violence overall.

While this effect is just from one set of simulations (again, see the Online Appendix for an illustration of how this relationship changes for random draws of the other parameters) and the changes in the average violence level are not large, this illustration has important empirical and policy consequences. First, it is interesting in light of empirical findings that competitive elections outside of consolidated democracies tend to be particularly violent (Hafner-Burton, Hyde, and Jablonski Reference Hafner-Burton, Hyde, Ryan and Jablonski2014; Straus and Taylor Reference Straus and Taylor2012), as well as the sharper spike of violence in elections where the chief executive is at stake shown in Figure 2.

Elections with higher stakes can be directly interpreted as ones where k_e is higher.Footnote ²⁰ Connecting the competitiveness of elections to this parameter is less straightforward. However, when elections are close there can be a higher marginal return to any political activity that will lead to more votes, including violence.Footnote ²¹ So, in that sense, we might expect that competitive elections are ones where the marginal returns to violence are high, which is precisely what we mean by k_e being high. Under this interpretation, it is the competitiveness of the election that depresses violence before the election: there is less of an incentive to commit violence today if the opposition has a real chance to take office in an upcoming election.

On the policy end, given that nearly every country holds some kind of election and few argue they should be abolished entirely, the comparison between more or less consequential elections is more relevant than the contrast between elections and no elections at all. Interestingly, this simulation suggests that to reap the benefits of elections in terms of peace it may not be enough to have elections that are only slightly consequential. Conversely, this example raises the possibility that making elections less consequential will lead to smaller spikes in violence during elections but also more violence overall.

Other Simulations

Figure 5 shows how changing several other parameters affects the difference in average levels of violence with and without elections. The left panel examines the effect of the discount rate. Changing δ has a nonmonotone effect. When the parties are extremely impatient (δ → 0) the optimal violence choice is always zero since the benefits of taking over office do not accrue right away. So if parties are completely impatient, there is no violence at all regardless of whether elections are held. For moderate levels of patience there can be more violence with elections, and the parties are patient enough to get a benefit from using violence but not patient enough for the prospect of future elections to outweigh the direct effect. When δ gets large, the indirect effect becomes stronger and elections lead to less violence overall.Footnote ²² For this particular parameterization, a discount rate of 0.88 is where elections go from promoting to preventing violence.

FIGURE 5.

Comparative Statics on Change in Violence with Elections by the Discount Rate (left panel), Value of Office (middle panel), and Frequence of Elections (right panel)

The middle panel examines the value of holding office (ψ ). For this parameterization (and, as shown in the Online Appendix, the majority but not all of our simulations), increasing ψ always leads to elections having a more pacifying effect. A potential explanation for this is that there will be more violence in general when there are more spoils to fight over. When there is already a high level violence in the nonelectoral baseline, the marginal effect of making violence more effective in electoral periods is relatively low compared to potential to decrease violence by making it more effective in the future.

Finally, the right panel examines the frequency of elections. When elections are extremely frequent, they lead to more violence, as there is not enough time between elections for the indirect effect of making nonelectoral periods more peaceful to outweigh the direct effect. When elections are very rare, they lead to slightly less violence since it takes a long time to wait until the next election, weakening the indirect effect. So, the optimal election timing is intermediate, where nonelectoral periods are short enough that the indirect effect matters for the entire time between elections, but long enough that the indirect effect adds up to matter more than the direct effect.

Bullets and Ballots

A central idea in the literature on democracy is that elections allow parties to compete for power with nonviolent means. Further, there are a number of empirical examples where armed groups have either delayed or renounced violence in the presence of nonviolent alternatives. For example, the Irish Republican Army (IRA) agreed to give up arms in Northern Ireland in 2005 when its political arm, Sinn Fein, made inroads in Northern Ireland and in the Irish Republic (Lavery and Cowell Reference Lavery and Cowell2005). Similarly, the Free Aceh Movement (GAM) in Indonesia give up violence as part of an agreement that included Indonesia allowing regional political parties to participate in national elections (Simanjuntak Reference Simanjuntak2008).

To combine our argument with the canonical claim that elections reduce violence by providing an alternative means to contest for power, we present an extension where the opposition can also use nonviolent actions. In short, we find that this leads to a substitution away from violence that can make society substantially more peaceful even though there is still a cycle of violence around elections.

To formalize this, suppose that in an electoral period, the opposition party can also choose to take nonviolent actions to increase their probability of becoming the incumbent following the election. Let x_t represent the amount of nonviolent activity in period t, which could represent campaigning, mobilizing voters on election day, or other less desirable but nonviolent tactics like vote buying. Let the probability of taking office in an electoral period be p(x_t + v_t ; k_t ) with the same functional form assumptions as above.Footnote ²³ So, now the k_t parameter reflects the increased effectiveness of political activity in general. To simplify the analysis, we assume that nonviolent political action is not available or ineffective in nonelectoral periods.

In addition to the cost c(v_t ) paid to take violent actions, the opposition also pays a partial cost c(x_t )/β _t for the nonviolent actions, for some β _t > 0. So, β _t reflects the relative cost of violent political activity in period t: when β _t > 1 violent activity is more efficient and when β _t < 1 nonviolent action is more efficient.Footnote ²⁴

Our solution concept (formalized in the Appendix) is analogous to the main model, except now the joint choice of violent and nonviolent action must be sequentially rational. Following standard optimization procedures, the first order condition for the opposition’s choice in period t is now the (v_t, x_t ) that jointly solve

(7) $$\begin{eqnarray} \delta \frac{\partial\, p(x_t + v_t; k_t)}{\partial v_t} \Delta ^{*}_{t+1} = c^\prime (v_t), \end{eqnarray}$$

(8) $$\begin{eqnarray} \delta \frac{\partial\, p(x_t + v_t; k_t)}{\partial x_t} \Delta ^{*}_{t+1} = c^\prime (x_t) / \beta_t , \end{eqnarray}$$

where Δ* _{t + 1} is again the difference between the net present value of entering period t + 1 as the incumbent versus opposition. This implies c′(v* _t ) = c′(x_t *)/β _t , which by the convexity of c ensures a unique pair (x* _t , v_t *) meeting these conditions. So, by an identical argument to the baseline model, there is a unique equilibrium level of violence and nonviolent activity (v* _t , x_t *) in each period characterized by Equations (7) and (8) when Δ* _t > 0 and v* _t = x_t * = 0 when Δ* _t ⩾ 0.

Figure 6 illustrates the impact of nonviolent activity on how much violence is chosen in equilibrium, with c(v_t ) = −v ² _t and p(v_t + x_t ; k_t ) = k_t [1 − (1 + v_t + x_t )⁻¹]. As before, the flat line in the left panel is the level of violence with no elections. The dotted curve represents the level of violence chosen where nonviolent technology is very expensive, and the spike in violence is large during the election is large relative to the dip in violence in no-electoral periods, resulting in more violence on average with elections despite some substitution to nonviolent activity.

FIGURE 6.

The Effect of Introducing Nonviolent Technologies on the Level of Violence

In the solid curve, the nonviolent technology is cheaper (high β _t , set to a constant β for all electoral periods), leading to a stronger substitution away from using violence during electoral periods. The availability of effective nonviolent technology also augments the intertemporal effects of the baseline model, as there is less reason to commit violence when it is possible to contest for power peacefully in future electoral periods. As a result, the level of violence is not only lower on average, but lower in every period than it would be without elections. The substitution effect is so large that violence peaks after the election, when (1) the nonviolent technology is not available, but (2) the next election is far enough away that it is worth committing violence to try and take over office. This pattern matches the case of Burkina Faso in Figure 1; we revisit the relationship between this example and several theoretical and empirical papers on postelection protest in the discussion. Still, the presence of elections induces a cyclical pattern of violence that, without thinking of the proper counterfactual, could erroneously lead to conclusions that elections encourage violence.

The right panel of Figure 6 plots the average level of violence as a function of the relative cost of violent actions (β). Unlike the previous graphs, making the nonviolent action cheaper (higher β) has an unambiguous and large effect in decreasing the average levels of violence. In this parameterization, elections would lead to more violence in the absence of a nonviolent technology (β → 0), but as long as the nonviolent technology is less than twice as expensive as violence the presence of elections will decrease the average level of violence.

Proof. See the Appendix. $\Box$

In other words, with substitution to non-violent technology the direct effect in the baseline model is weakened if not reversed, potentially leading to less violence in electoral periods. The logic of the indirect effect of reducing violence in non-electoral periods remains, and so elections can unambiguously reduce violence on average and even in every period (as in Figure 6).

Incumbent and Opposition Violence

Next, we analyze a case where both the incumbent and opposition can commit violence. In this section, we assume the probability that the opposition becomes the incumbent in period t + 1 as a function of their violence choice (now v _{O, t} ⩾ 0) and the incumbent violence choice (v _{I, t} ⩾ 0) is

$$\begin{eqnarray*} p(v_{O,t}, v_{I, t}; k_t) = \left\lbrace \begin{array}{@{}l@{\quad }l@{}}k_t \frac{v_{O,t}}{v_{O,t} + v_{I,t}} & v_{O,t} + v_{I,t} > 0\\ p_0 & v_{O,t} + v_{I,t} = 0 \end{array}\right. \end{eqnarray*}$$

for some k_t ∈ [0, 1] and p ₀ ∈ (0, 1). (The v _{O, t} + v _{I, t} = 0 case is to avoid division by zero, and the exact value of p ₀ has no impact on the analysis.) This functional form meets all of the assumptions with respect to v _{O, t} and k_t in the baseline model as long as v _{I, t} > 0 (which will always be true in equilibrium).

Let the opposition cost for committing violence level v _{O, t} be exactly v _{O, t} , and let the cost of committing violence for the incumbent be γv _{I, t} for some 0 < γ < 1. That is, we assume that violence is “cheaper” for the incumbent, or, equivalently, the incumbent gets a higher return to violence.Footnote ²⁵

The rents for being the incumbent are again ψ. So, actor J’s net present value of committing violence level v _{J, t} in period t (given the future violence choices) is

$$\begin{eqnarray*} \pi _O(t, v_{O,t}; v_{I, t}) &=& -v_{O,t} + \delta [p(v_{O,t}, v_{I,t}; k_t) \pi _I^{*}(t+1)\\ &&+\, [1 - p(v_{O,t}, v_{I, t}; k_t)] \pi _O^{*}(t+1)],\\ \pi _I(t, v_{O,t}; v_{I, t}) &=& \psi\; - \gamma v_{I,t} + \delta [p(v_{O,t}, v_{I,t}; k_t) \pi _O^{*}(t+1)\\ &&+\, [1 - p(v_{O,t}, v_{I, t}; k_t)] \pi _I^{*}(t+1)]. \end{eqnarray*}$$

Our solution concept now requires that the violence choices are mutual best responses given future violence choices (see the Appendix for a formal statement) By a standard analysis, whenever π* _I (t + 1) > π _O *(t + 1) the equilibrium violence choices are given by:

(9) $$\begin{eqnarray} v_{O,t}^{*} = \frac{k_t \delta [\pi _I^{*}(t+1) - \pi _O^{*}(t+1)]}{(1 + \gamma ^{-1})^2}, \end{eqnarray}$$

(10) $$\begin{eqnarray} v_{I,t}^{*} = \frac{\gamma ^{-1} k_t \delta [\pi _I^{*}(t+1) - \pi _O^{*}(t+1)]}{(1 + \gamma ^{-1})^2} \end{eqnarray}$$

and if π* _I (t + 1) < π _O *(t + 1) both actors choose no violence (see the Appendix for detail). Since π* _I (T + 1) > π _O *(T + 1), the level of violence chosen in the last period is positive for both actors: v* _{J, T} > 0.

It immediately follows from equations (9) and (10) that the level of violence chosen by both actors in period t (when strictly positive) is increasing in the effectiveness in period t. Further, since γ < 1, the incumbent always chooses more violence than the opposition, and hence always remains the incumbent with a probability greater than 1/2. As γ → 0, the incumbent commits far more violence than the opposition, indicating this extension can capture cases where the vast majority is committed by government forces.

Using these violence levels to compute the net present value for entering period t as each party, the difference between these payoffs is

(11) $$\begin{eqnarray} \Delta ^{*}_t = \pi _I^{*}(t) - \pi _O^{*}(t) = \psi\, + \delta \left(1 - 2 \frac{k_t}{1 + \gamma ^{-1}}\right) \Delta ^{*}_{t+1}. \nonumber\\ \end{eqnarray}$$

Since γ < 1, whenever Δ* _{t + 1} is positive, Δ* _t is positive as well, so by induction Δ* _t is positive for all t and hence both actors choose a strictly positive level of violence in each period. Further, Δ* _t is decreasing in k_t : when violence is more effective in period t, the difference between entering this period as the incumbent versus opposition decreases. So, there will be less violence committed in period t − 1 as k_t increases.

The same inductive argument then implies that Δ* _{t − 1} is decreasing in k_t , as is Δ* _t′ for any t′ < t, and hence v* _{J, t − 1} is decreasing in k_t for both J = I and J = O. Combined with the fact that the violence levels in period t are increasing in k_t :

Proof. See the Appendix. $\Box$

The intuition for the opposition violence choice is similar to the main model: if violence will be more effective in the future (e.g., because of an election), the opposition invests less in violence anticipating it will be more effective later. The fact that violence will be more effective in the future also leads the incumbent to choose less, as even if they are ousted they will have an opportunity to take power back during the election.

Another intuition for the incumbent violence choice is that they are reacting to anticipated violence choices by the opposition, and increasing the violence choice by the opposition gives the incumbent an incentive to commit more violence to counteract this. Formally, the marginal benefit for the incumbent violence choice is increasing in v _{O, t} as long as v _{O, t} < v _{I, t} , which is always true in equilibrium. That is, making violence more effective for the opposition during elections can lead to more incumbent violence. During elections, this can lead to a spiral effect where the incumbent reacts to opposition violence with even more of their own. However, there is a similar but peaceful spiral effect in nonelectoral periods: since the opposition would rather wait to use violence, the government has less reason to repress them.

Continuous Policy Outcomes

Finally, a drawback of the models so far is that they only consider violence committed to take control of the government, while much political violence has more incremental goals. To show that the logic of our model applies to this more general class of political violence, we briefly present a variant of the model where a single political actor chooses a level of violence to affect a continuous policy outcome. Unlike before, there is only one actor and no “incumbency”; i.e., the same actor chooses a violence level in each period to influence the policy in their preferred direction.

Formally, the actor chooses violence level v_t in each of T > 1 periods. The policy in period t + 1 is

$$\begin{eqnarray*} y_{t+1} = (1 - k_t) y_{t} + k_t v_t, \end{eqnarray*}$$

where k_t ∈ [0, 1]. That is, the policy in each period is a weighted average of the past policy and the efforts of the political actor. When k_t is higher, violence will be more effective in period t, or, equivalently, the status quo is “less sticky.” The period utility as a function of the policy and violence choice is

$$\begin{eqnarray*} u_t = y_t - c v_t^2 \end{eqnarray*}$$

and the payoff for the entire game is

$$\begin{eqnarray*} \sum _{i=1}^T \delta ^{i-1} u_i + a \delta ^T y_{T+1}, \end{eqnarray*}$$

where a > 0 represents the relative importance of the policy at the end of the game.

The main insight from this model is that violence committed in period t increases the policy in every future period. However, the impact on periods beyond the next one is diminished by the fact that future policies are also shaped by future violence choices. In particular, a marginal increase in violence in period t increases y _{t + 1} by k_t, y _{t + 2} by k_t (1 − k _{t + 1}), and for any τ > t + 1, y _τ by k_t ∏^{τ − 1} _{i = t + 1}(1 − k_i ). So, the total return to violence today is increasing in k_t and decreasing in the effectiveness of violence in every future period:

In addition to demonstrating that the results in the baseline model extend to “policies” other than who controls the government, this model provides an alternative intuition for why elections may reduce violence overall. When committing violence today, political actors are not just attempting to change the status quo for the “next period,” whether that be a day, year, or electoral cycle. Since policy tends to be sticky, the marginal benefit to violence today is a discounted sum of how violence today affects policy for every future period. The effective discount rate for the future is not just a function of patience, but whether whatever political gains made today could be reversed in the future. So, if elections increase the chance that any political gains may be reversed, there is less reason to commit violence in nonelectoral periods.